Meusburger C, Wise D (2021)
Publication Type: Journal article
Publication year: 2021
Article Number: 2150016
DOI: 10.1142/S0129055X21500161
We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearized analogs. Starting from physical considerations, we derive an axiomatic definition of Hopf algebra gauge theory, including locality conditions under which the theory for a general ribbon graph can be assembled from local data in the neighborhood of each vertex. For a vertex neighborhood with n incoming edge ends, the algebra of non-commutative 'functions' of connections is dual to a two-sided twist deformation of the n-fold tensor power of the gauge Hopf algebra. We show these algebras assemble to give an algebra of functions and gauge-invariant subalgebra of 'observables' that coincide with those obtained in the combinatorial quantization of Chern-Simons theory, thus providing an axiomatic derivation of the latter. We then discuss holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this gives, for each path in the embedded graph, a map from connections into the gauge Hopf algebra, depending functorially on the path. Curvatures-holonomies around the faces canonically associated to the ribbon graph-then correspond to central elements of the algebra of observables, and define a set of commuting projectors onto the subalgebra of observables on flat connections. The algebras of observables for all connections or for flat connections are topological invariants, depending only on the topology, respectively, of the punctured or closed surface canonically obtained by gluing annuli or discs along edges of the ribbon graph.
APA:
Meusburger, C., & Wise, D. (2021). Hopf algebra gauge theory on a ribbon graph. Reviews in Mathematical Physics. https://dx.doi.org/10.1142/S0129055X21500161
MLA:
Meusburger, Cathérine, and Derek Wise. "Hopf algebra gauge theory on a ribbon graph." Reviews in Mathematical Physics (2021).
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